User:Lériendil/Third-superparticulars and semiparticulars by prime subgroup
Some shorthand notation used here:
- Sk stands for k^2/[(k-1)(k+1)] by standard convention (the kth square superparticular).
- Gk stands for S(k-1)*Sk*S(k+1) (the kth third-particular).
- Rk stands for S(k-1)/S(k+1) (the kth semiparticular).
- Tk = Sk * S(k+1) stands for [k(k+1)/2]/[(k-1)(k+2)/2] (the kth triangle superparticular).
- Lp refers to the p-limit, i.e. the subgroup of primes less than or equal to p.
- Lp(-q) refers to the p limit with the prime q omitted: e.g. L17(-11) refers to the 2.3.5.7.13.17 subgroup; these omissions can be stacked so that L23(-5.17) refers to the group 2.3.7.11.13.19.23.
Note that not all members of Gk and Rk are superparticular. In particular, G(3k) is throdd-particular, and R(4k) is odd-particular. Such ratios will be excluded from consideration in this chart, though they will appear on companion no-twos and no-threes pages.
This list eventually aims to be complete to the 29-add-one-limit, i.e. the class of subgroups with at most one prime greater than 29, which is a superset of the 31-limit.
2- and 3-prime subgroups (2.p and 2.3.p)
| Third-particular | Subgroup | Comma | Semiparticular | Subgroup | Comma | ||
|---|---|---|---|---|---|---|---|
| Ratio | Smonzo | Ratio | Smonzo | ||||
| R3 = G4 | 2.5 | 5/4 | [-2 1⟩ | ||||
| G4 = R3 | 2.5 | 5/4 | [-2 1⟩ | ||||
| G7 = S4 | L5 | 16/15 | [4 -1 -1⟩ | R7 = S9 | L5 | 81/80 | [-4 4 -1⟩ |
| G5 | 2.7 | 8/7 | [3 -1⟩ | R5 = T7 | 2.3.7 | 28/27 | [2 -3 1⟩ |
| G10 | 2.3.11 | 33/32 | [-5 1 1⟩ | R10 | 2.3.11 | 243/242 | [-1 5 -2⟩ |